Bianchi IX Cosmologies, Spectral Action, and Modular Forms

Transcription

1 Bianchi IX Cosmologies, Spectral Action, and Modular Forms Matilde Marcolli, Ma148 Spring 2016 Topics in Mathematical Physics

2 References: Wentao Fan, Farzad Fathizadeh, Matilde Marcolli, Spectral Action for Bianchi type-ix cosmological models, arxiv: , J. High Energy Phys. (2015) 85, 28 pp. Wentao Fan, Farzad Fathizadeh, Matilde Marcolli, Modular forms in the spectral action of Bianchi IX gravitational instantons, arxiv: Based on Wentao s SURF project with Farzad

3 SU(2)-Bianchi IX cosmologies (Euclidean, compactified) another version of Bianchi IX mixmaster cosmologies, with SU(2) symmetry (Euclidean version) g = w 1 w 2 w 3 dt 2 + w 2w 3 w 1 σ w 1w 3 w 2 σ w 1w 2 w 3 σ 2 3 with w i = w i (t), or more generally g = F (dµ 2 + σ2 1 w σ2 2 w σ2 3 w3 3 ) with a conformal factor F w 1 w 2 w 3 SU(2) invariant 1-forms {σ i } satisfying relations dσ i = σ j σ k for all cyclic permutations (i, j, k) of (1, 2, 3)

4 more explicitly ds 2 is + w 1 w 2 w 3 dt dt + w 1w 2 cos(η) ( w2 w 3 sin 2 (η) cos 2 (ψ) w 1 + (w 1 2 w 2 2 )w 3 sin(η) sin(ψ) cos(ψ) ( w 1 w 2 + w2 w 3 sin 2 (ψ) w 1 w 3 + w 1 ( w3 sin 2 (η) sin 2 (ψ) w 2 ) + w 1w 3 cos 2 (ψ) w 2 dφ dψ + w 1w 2 cos(η) w 3 dψ dφ + w 2 cos 2 (η) w 3 )) dφ dφ dη dφ + (w 1 2 w 2 2 )w 3 sin(η) sin(ψ) cos(ψ) w 1 w 2 dφ dη dη dη + w 1w 2 w 3 dψ dψ identifying S 3 with unit quaternions SU(2) The metrics on S 3 w 2 w 3 w 1 σ w 1w 3 w 2 σ w 1w 2 w 3 σ 2 3 are left-invariants under the action of SU(2) but not right-invariant (unlike the round metric on S 3 )

5 Spectral action models of gravity (modified gravity) Spectral triple: (A, H, D) 1 unital associative algebra A 2 represented as bounded operators on a Hilbert space H 3 Dirac operator: self-adjoint D = D with compact resolvent, with bounded commutators [D, a] prototype: (C (M), L 2 (M, S), /D M ) extends to non smooth objects (fractals) and noncommutative (NC tori, quantum groups, NC deformations, etc.)

6 Action functional Suppose finitely summable ST = (A, H, D) ζ D (s) = Tr( D s ) <, R(s) >> 0 Spectral action (Chamseddine Connes) S ST (Λ) = Tr(f (D/Λ)) = Mult(λ)f (λ/λ) λ Spec(D) f = smooth approximation to (even) cutoff

7 Asymptotic expansion (Chamseddine Connes) for (almost) commutative geometries: Tr(f (D/Λ)) β Σ + ST f β Λ β D β + f (0) ζ D (0) Residues D β = 1 2 Res s=β ζ D (s) Momenta f β = 0 f (v) v β 1 dv Dimension Spectrum Σ ST poles of zeta functions ζ a,d (s) = Tr(a D s ) positive dimension spectrum Σ + ST = Σ ST R +

8 Zeta function and heat kernel (manifolds) Mellin transform heat kernel expansion D s = 1 Γ(s/2) 0 e td2 t s 2 1 dt Tr(e td2 ) = α t α c α for t 0 zeta function expansion ζ D (s) = Tr( D s ) = α c α Γ(s/2)(α + s/2) + holomorphic taking residues Res s= 2α ζ D (s) = 2c α Γ( α)

9 Pseudo-differential Calculus: (manifold case) to obtain full asymptotic expansion of the Spectral Action Dirac operator D and pseudodifferential symbol of D 2 σ(d 2 )(x, ξ) = p 2 (x, ξ) + p 1 (x, ξ) + p 0 (x, ξ) each p k homogeneous of order k in ξ Cauchy integral formula e td2 = 1 2πi γ e tλ (D 2 λ) 1 dλ Seeley de-witt coefficients (m = dim M) Tr(e td2 ) t 0+ t m/2 a 2n (D 2 )t n n=0

10 Parametrix Method D 2 order 2 elliptic differential operator: exists a parametrix R λ with σ(r λ ) r j (x, ξ, λ) j=0 r j (x, ξ, λ) pseudodifferential symbol order 2 j r j (x, tξ, t 2 λ) = t 2 j r j (x, ξ, λ) R λ approximates (D 2 λ) 1 with σ((d 2 λ)r λ ) 1 recursive equation: σ((d 2 λ)r λ ) ((p 2 (x, ξ) λ) + p 1 (x, ξ) + p 0 (x, ξ)) r j (x, ξ, λ) 1 j=0

11 solution for R λ constructed recursively: r 0 (x, ξ, λ) = (p 2 (x, ξ) λ) 1 r n (x, ξ, λ) = 1 α! α ξ r j(x, ξ, λ) D α x p k (x, ξ) r 0 (x, ξ, λ), summation over all α Z 4 0, j {0, 1,..., n 1}, k {0, 1, 2}, with α + j + 2 k = n Seeley-deWitt coefficients and Parametrix Method a 2n (x, D 2 ) = (2π) m e λ tr (r 2n (x, ξ, λ)) dλ d m ξ 2πi odd j coefficients vanish: r j (x, ξ, λ) odd function of ξ γ

12 Dirac operator orthonormal coframe {θ a } D = a θ a S θ a spin connection S with matrix of 1-forms ω = (ω a b ) with θ a = b ω a b θb metric-compatibility and torsion-freeness (Levi Civita connection) ω a b = ωb a, dθ a = b ω a b θb

13 Dirac operator D = a,µ with ω b a = c ωb acθ c γ a dx µ (θ a ) x µ + 1 γ c ω 4 acγ b a γ b a,b,c matrices γ a Clifford action of θ a on spin bundle: (γ a ) 2 = I γ a γ b + γ b γ a = 0 for a b

14 Dirac operator on Bianchi IX metrics local coordinates (x µ ) = (t, η, φ, ψ) with S 3 parametrized by (η, φ, ψ) (cos(η/2)e i(φ+ψ)/2, sin(η/2)e i(φ ψ)/2) with 0 η π, 0 φ < 2π, 0 ψ < 4π. orthonormal frame θ 0 = w 1 w 2 w 3 dt, θ 1 w2 w 3 = sin(η) cos(ψ) w 1 θ 2 w1 w 3 = sin(η) sin(ψ) w 2 w2 w 3 dφ sin(ψ) w 1 w1 w 3 dφ + cos(ψ) θ 3 w1 w 2 w1 w 2 = cos(η) dφ + dψ. w 3 w 3 w 2 dη, dη,

15 non-vanishing ω b ac ω11 0 = w 2 (w 1 w 3 w 3w 1 ) + w 1w 3 w 2, ω 0 2(w 1 w 2 w 3 ) 3/2 22 = w 2 (w 3 w 1 + w 1w 3 ) w 1w 3 w 2, 2(w 1 w 2 w 3 ) 3/2 ω33 0 = w 2 (w 3 w 1 w 1w 3 ) + w 1w 3 w 2, ω 1 2(w 1 w 2 w 3 ) 3/2 23 = w 1 2w 2 2 w ( ) 3 2 w w2 2, 2(w 1 w 2 w 3 ) 3/2 ω32 1 = w ( ) 1 2 w 2 2 w3 2 + w 2 2 w3 2, ω 2 2(w 1 w 2 w 3 ) 3/2 31 = w 2 2w 3 2 w ( ) 1 2 w w3 2. 2(w 1 w 2 w 3 ) 3/2

16 pseudo-differential symbol of Dirac σ(d)(x, ξ) = iγ a e µ 1 a ξ µ w a,µ 1 w 2 w 3 ( w1 w 2 w w w w3 2 ( w 1 + w 2 w 1 ) γ 2 γ 3 γ 4 3 w 3 w 2 + w = iγ2 w 1 (csc(η) cos(ψ) (ξ 4 cos(η) ξ 3 ) + ξ 2 sin(ψ)) w2 w3 + iγ3 w 2 (sin(ψ) (ξ 3 csc(η) ξ 4 cot(η)) + ξ 2 cos(ψ)) w1 w3 iγ 1 ξ iγ4 ξ 4 w3 w1 w2 w3 w1 w2 ( ) w 1 + w 2 + w 3 γ 1 w 1 w 2 w 3 w 1 w 2 w 3 ( w1 w 2 w w w ) w3 2 γ 2 γ 3 γ 4 ) γ 1

17 with non-vanishing e µ a : e 0 0 = 1 w1 w 2 w 3, e1 1 w1 sin(ψ) =, w2 w 3 e 1 2 = e 2 2 = w2 cos(ψ), e 2 w1 csc(η) cos(ψ) 1 =, w1 w 3 w2 w 3 w2 csc(η) sin(ψ), e 3 w1 cot(η) cos(ψ) 1 =, w1 w 3 w2 w 3 e2 3 w2 cot(η) sin(ψ) =, e3 3 = w1 w 3 w3 w1 w 2

18 get from the symbol the homogeneous components p k (x, ξ) with σ(d 2 )(x, ξ) = p 2 (x, ξ) + p 1 (x, ξ) + p 0 (x, ξ) Example: for p 0 (x, ξ) get ( w 1 8w 1 w 2 2 w 1 8w 1 w w 2 8w w 1 8w 3 1 ( w 1 4w1 2w + w 1 w 2 2w 3 8w1 2w 2 2w + w 1 w w 2 w 3 8w 1 w 2 2 w w 3 8w 1 w 2 + w 1w 3 16w 3 2 w 2 8w1 2w w 3 2 8w1 2w w 2 3 8w 2 w3 2 )γ 1 γ 2 γ 3 γ 4 + w 3 8w2 2w + 3w 3 3 8w3 3 5w 2 8w1 2w + 2w w1 3w w 2 2w 3 4w 1 w2 2w 3 5w w 1 w2 3w w 3 5w w 1 w 2 w w 2w 3 16w 1 w 2 w3 3 16w1 3 + w 2 8w 1 w 3 + w 1 8w 2 w 3 + w 1w 2 )I. 16w3 3 also manageable expression for p 2 (x, ξ), longer one for p 1 (x, ξ)

19 Applying Parametrix Method to this Dirac operator a 2n (x, D 2 ) = (2π) m e λ tr (r 2n (x, ξ, λ)) dλ d m ξ 2πi Find a 0, a 2, a 4 explicitly γ a 0 (D 2 ) = 4w 1 w 2 w 3 a 2 (D 2 ) = w w w w 2 1 w 2 2 6w 2 3 (w 3 ) 2 6w w 2 1 w 2 3 6w w 2 2 w 2 3 6w 2 1 (w 1 ) 2 6w 2 1 (w 2 ) 2 6w 2 2 w 1 w 2 w 1 w 3 w 2 w 3 + w 1 + w 2 + w 3. 3w 1 w 2 3w 1 w 3 3w 2 w 3 3w 1 3w 2 3w 3 and a much longer and more complicated expression for a 4 (D 2 ) Observation: all coefficients in these expressions (also for a 4 ) are rational numbers... what about other terms in expansion?

20 A different method: Wodzicki residue Wodzicki residue: unique trace functional on algebra of pseudodifferential operators on smooth sections of vector bundle over smooth manifold classical pseudodifferential operator P σ of order d Z local symbol σ(x, ξ) σ d j (x, ξ) (ξ ), j=0 σ d j positively homogeneous order d j in ξ Residue: Res(P σ ) = S M Tr (σ m (x, ξ)) d m 1 ξ d m x, S M = {(x, ξ) T M; ξ g = 1} cosphere bundle

21 spectral formulation of residue: pseudodifferential operator P σ, Laplacian P σ Res s=0 Tr(P σ s ) same up to a constant c m = 2 m+1 π m Mellin transform (for simplicity Ker( ) = 0): heat kernel expansion Tr( s ) = 1 Γ(s) Tr ( e t ) = t m/2 N n=0 0 Tr(e t ) t s dt t a 2n t n + O(t m/2+n+1 )

22 find for any non-negative integer n m/2 1: Res s=m/2 n Tr( s ) = a 2n( ) Γ(m/2 n), in particular Res s=1 Tr( s ) = a m 2 ( ) in terms of Wodicki residue: a m 2 ( ) = 1 c m Res( 1 ) = 1 2 m+1 π m Res( 1 )

23 applied to = D 2 coefficient a 2 (D 2 ) a 2 (D 2 ) = 1 c 4 Res(D 2 ) = 1 32π 4 S M Tr ( σ 4 (D 2 ) ) d 3 ξ d 4 x for other coefficients, introduce an auxiliary product space for correct counting of dimensions: use flat r-dimensional torus T r = (R/Z) r = D T r, T r flat Laplacian on T r a 2+r (D 2 ) = because Künneth formula gives π 4+r/2 Res( 1 ) a 2+r ((x, x ), ) = a 2+r (x, D 2 )a 0 (x, T r ) = 2 r π r/2 a 2+r (x, D 2 ) with volume term only non-zero heat coefficient for flat metric

24 obtain for all coefficients a 2+r (D 2 ) = π 4+r/2 Tr ( σ 4 r ( 1 ) ) d 3+r ξ d 4 x. writing σ( 1 ) j= 2 σ j(x, ξ) inductively σ 2 (x, ξ) = p 2(x, ξ) 1, σ 2 n (x, ξ) = 1 α! α ξ σ j(x, ξ) D α x p k (x, ξ) σ 2 (x, ξ) (n > 0), summation over all multi-indices non-negative integers α, 2 n < j 2, 0 k 2, with α j k = n

25 setting ζ µ+1 = ν eν µξ ν+1 find inductively for n 2 σ 2 n (x, ξ) S (M T n 2 ) = σ 2 n(x, ξ(ζ)) ζ S n+1 = (w 1 w 2 w 3 ) 3 2 n P n (ζ) polynomials P n (ζ) coefficients functions of w i and derivatives these explicitly give ( ) a 2n (D 2 ) = (w 1 w 2 w 3 ) 1 3n Q 2n w 1, w 2, w 3, w 1, w 2, w 3,..., w (2n) 1, w (2n) 2, w (2n) 3 with Q 2n polynomials with rational coefficients Q 2n = 1 2π n+1 Tr(P 2n (ζ)( 1 )) d 2n+1 ζ S 2n+1 Question: is this rationality a sign of an arithmetic structure of Bianchi IX gravitational instantons that persists in the Spectral Action?

26 BIanchi IX gravitational instantons and Painlevé VI Euclidean Bianchi IX metrics with SU(2)-symmetry that are - self-dual (Weyl curvature tensor W self-dual) - Einstein metrics (Ricci tensor proportional to the metric) Self-dual equations for a Riemannian 4-manifold are PDEs; with SU(2)-symmetry reduce to ODEs This ODE is a Painlevé VI equation with (α, β, γ, δ) = ( 1 8, 1 8, 1 8, 3 8 )

27 N.J. Hitchin. Twistor spaces, Einstein metrics and isomonodromic deformations, J.Diff.Geom., Vol. 42, No. 1 (1995), K.P. Tod. Self dual Einstein metrics from the Painlevé VI equation, Phys. Lett. A 190 (1994), S. Okumura. The self dual Einstein Weyl metric and classical solutions of Painlevé VI, Lett. in Math. Phys., 46 (1998), M.V. Babich, D.A. Korotkin, Self dual SU(2) Invariant Einstein Metrics and Modular Dependence of Theta Functions. Lett. Math. Phys. 46 (1998),

28 Painlevé VI equations Painlevé transcendents: solutions of nonlinear second-order ODEs in the plane with Painlevé property (the only movable singularities are poles) not solvable in terms of elementary functions; classification in types Painlevé VI: 4-parameter family (α, β, γ, δ) d 2 X dt 2 = 1 ( 1 2 X + 1 X ) ( ) dx 2 X t dt X (X 1)(X t) + t 2 (t 1) 2 ( 1 t + 1 ) dx dt + t X t ( α + β t X 2 + γ t 1 t(t 1) + δ (X 1) 2 (X t) 2 ).

29 Painlevé VI and elliptic curves Painlevé VI rewritten as (Fuchs) t(1 t) [t(1 t) d 2 dt 2 + (1 2t) d dt 1 ] (X,Y ) dx = 4 x(x 1)(x t) = αy + β ty (t 1)Y + γ X 2 (X 1) 2 + (δ 1 1)Y )t(t 2 (X t) 2 where (X, Y ) := (X (t), Y (t)) is a section (local and/or multivalued) P := (X (t), Y (t)) of the generic elliptic curve E = E(t) : Y 2 = X (X 1)(X t) left-hand-side µ(p) satisfies µ(p + Q) = µ(p) + µ(q) for P + Q addition on the elliptic curve E (in particular µ(q) = 0 for points of finite order)

30 analytic description of the elliptic curve E τ = C/Λ with Λ = Z + τz, with τ H then Painlevé VI rewritten as (Manin) d 2 z dτ 2 = 1 (2πi) 2 3 j=0 α j z (z + T j 2, τ) with (α 0,..., α 3 ) := (α, β, γ, 1 2 δ) and (T 0, T 1, T 2, T 3 ) := (0, 1, τ, 1 + τ), and (z, τ) := 1 z 2 + ( ) 1 (z mτ n) 2 1 (mτ + n) 2 (m,n) (0,0)

31 also have, for e i (τ) = ( T i 2, τ) z (z, τ) 2 = 4( (z, τ) e 1 (τ))( (z, τ) e 2 (τ))( (z, τ) e 3 (τ)) so e 1 + e 2 + e 3 = 0 a multivalued solution z = z(τ) defines a multi section of the family, which is a covering of H is know ramification and monodromy can study behavior over geodesics in H - Yu.I. Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of P 2, in Geometry of Differential Equations, Amer. Math. Soc. Transl. (2) Vol. 186 (1998)

32 Theta characteristics explicit paremeterization of solutions for coefficients W i of the Bianchi IX gravitational instantons (from solutions of Painlevé VI) theta characteristics with parameters (p, q): ϑ[p, q](z, iµ) := m Z exp ( π(m + p) 2 µ + 2πi(m + p)(z + q) ) theta-characteristics and theta functions with vanishing characteristics ϑ[p, q](z, iµ) = exp ( πp 2 µ + 2πipq ) ϑ[0, 0](z + piµ + q, iµ)

33 Gravitational instantons and theta characteristics use notation ϑ[p, q] := ϑ[p, q](0, iµ), and ϑ 2 := ϑ[1/2, 0], ϑ 3 := ϑ[0, 0], ϑ 4 := ϑ[0, 1/2] self-dual metrics with g = F (dµ 2 + σ2 1 w σ2 2 w σ2 3 w3 3 ) w 1 = i 2 ϑ q 3ϑ ϑ[p, q ] 4 e πip, w 2 = i ϑ[p, q] 2 ϑ q 2ϑ ϑ[p + 1 2, q ] 4 e πip, ϑ[p, q] w 3 = 1 2 ϑ q 2ϑ ϑ[p + 1 2, q] 3, ϑ[p, q] with non-zero cosmological constant Λ: F = 2 πλ w 1 w 2 w 3 ( q log ϑ[p, q])2

34 these metrics also satisfy Einstein equation if either 1 Λ < 0 with p R and q ir 2 Λ > 0 with q R and p ir also case with vanishing cosmological constant: w 1 = 1 µ + q d dµ log ϑ 2, w 2 = 1 µ + q d dµ log ϑ 3, w 3 = 1 µ + q d dµ log ϑ 4, F = C(µ + q 0 ) 2 w 1 w 2 w 3 with q 0, C R, C > 0. - M.V. Babich, D.A. Korotkin, Self dual SU(2) Invariant Einstein Metrics and Modular Dependence of Theta Functions. Lett. Math. Phys. 46 (1998), Yuri Manin, Matilde Marcolli, Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies, arxiv: [gr-qc]

35 Bianchi IX: time-dependent conformal perturbations original triaxial Bianchi IX: ds 2 = w 1 w 2 w 3 dµ 2 + w 2w 3 w 1 σ w 3w 1 w 2 σ w 1w 2 w 3 σ 2 3 w i = w i (µ) cosmic time µ time-dependent conformal perturbation: ( d s 2 = Fds 2 = F w 1 w 2 w 3 dµ 2 + w 2w 3 σ1 2 + w 3w 1 σ2 2 + w ) 1w 2 σ3 2 w 1 w 2 w 3 with F = F (µ) effect on Dirac operator: D = 1 3F D + γ 0 F 4F 3 2 w 1 w 2 w 3 D Dirac operator of unperturbed Bianchi IX

36 spectral action expansion for D from heat kernel ( ) Tr exp( t D 2 ) t 2 ã 2n t n, t 0 + n=0 rationality result for coefficients of the spectral action ã 2n = Q 2n (w 1, w 2, w 3, F, w 1, w 2, w 3, F,..., w (2n) 1, w (2n) 2, w (2n) 3, F (2n) ) F 2n (w 1 w 2 w 3 ) 3n 1 Q 2n polynomial with rational coefficients

37 zeroth coefficient: volume form (cosmological term) ã 0 = 4F 2 w 1 w 2 w 3 second coefficient ã 2 : Einstein-Hilbert action F 3 F 3 ( ) w1 2 +w2 2 +w3 2 + F 6 ( w 2 1 w 2 2 w 2 3 w w1 2w3 2 w 2 2 w2 2 ( w 1 w 2 + w 1 w 3 + w 2 w ) 3 + F ( w 1 + w 2 + w 3 w 1 w 2 w 1 w 3 w 2 w 3 3 w 1 w 2 w 3 ) much longer and more complicated explicit formula for ã 4 (Weyl conformal gravity and Gauss-Bonnet gravity) + w2 2w3 2 w 2 ) 1 w1 2 F 2 2F + F

38 Gravitational Instantons now assuming conformally perturbed Bianchi IX is self-dual Einstin metric and use parameterization by theta functions two-parameter family with non-vanishing cosmological constant: w 1 [p, q](iµ) = i 2 ϑ 3(iµ)ϑ 4 (iµ) qϑ[p, q ](iµ) e πip ϑ[p, q](iµ) w 2 [p, q](iµ) = i 2 ϑ 2(iµ)ϑ 4 (iµ) qϑ[p + 1 2, q ](iµ) e πip ϑ[p, q](iµ) w 3 [p, q](iµ) = 1 2 ϑ 2(iµ)ϑ 3 (iµ) qϑ[p + 1 2, q](iµ) ϑ[p, q](iµ) F [p, q](iµ) = 2 1 πλ ( q ln ϑ[p, q](iµ)) 2 = 2 ( ϑ[p, q](iµ) πλ q ϑ[p, q](iµ) ) 2

39 one-parameter family with vanishing cosmological constant: w 1 [q 0 ](iµ) = w 2 [q 0 ](iµ) = w 3 [q 0 ](iµ) = F [q 0 ](iµ) = C(µ + q 0 ) 2, C arbitrary positive constant d µ + q 0 dµ log ϑ 2(iµ), d µ + q 0 dµ log ϑ 3(iµ), d µ + q 0 dµ log ϑ 4(iµ),

40 Modularity generators of the modular group PSL 2 (Z) T 1 (τ) = τ + 1, S(τ) = 1 τ, τ H using behavior of theta functions and derivatives under modular transformations (two-parameter family):

41 for µ with R(µ) > 0: similar results for w 2 and w 3 under modular generator S

42 conformal factor: F [p, q](iµ + 1) = F [p, q + p ](iµ), F (n) [p, q](iµ + 1) = F (n) [p, q + p ](iµ). F [p, q]( i µ ) = µ 2 F [ q, p](iµ), F [p, q]( i µ ) = F [ q, p](iµ) 2µ 1 F [ q, p](iµ), F [p, q]( i µ ) = µ2 F [ q, p](iµ) + 2µF [ q, p](iµ) 2F [ q, p](iµ), F (3) [p, q]( i µ ) = µ4 F (3) [ q, p](iµ), F (4) [p, q]( i µ ) = µ6 F (4) [ q, p](iµ) 4µ 5 F (3) [ q, p](iµ).

43 similar results for the case of the one-parameter family with vanishing cosmological constant modularity of spectral action coefficients: ã 0 [p, q](iµ + 1) = ã 0 [p, q + p ](iµ) ã 2 [p, q](iµ + 1) = a 2 [p, q + p ](iµ) ã 4 [p, q](iµ + 1) = ã 4 [p, q + p ](iµ) ã 0 [p, q]( i µ ) = µ2 ã 0 [ q, p](iµ) ã 2 [p, q]( i µ ) = µ2 ã 2 [ q, p](iµ) ã 4 [p, q]( i µ ) = µ2 ã 4 [ q, p](iµ)

44 Modularity of remaining coefficients ã 2n Dirac operators D 2 [p, q], D 2 [p, q + p ] and D 2 [ q, p] are isospectral ( ) heat kernel K t [p, q] of exp t D 2 [p, q] in terms of eigenvalues and eigenspinors modularity K t [p, q](iµ 1 + 1, iµ 2 + 1) = K t [p, q + p ](iµ 1, iµ 2 ), K t [p, q]( 1 iµ 1, 1 iµ 2 ) = (iµ 2 ) 2 K t [ q, p](iµ 1, iµ 2 ). then modularity of coefficients ã 2n : ã 2n [p, q](iµ + 1) = ã 2n [p, q + p ](iµ), ã 2n [p, q]( i µ ) = (iµ)2 ã 2n [ q, p](iµ).

45 Vector valued modular forms coefficients satisfy: ã 2n [p + 1, q] = ã 2n [p, q + 1] = ã 2n [p, q], PSL 2 (Z) action on (p, q) R/Z 2 : S(p, q) = ( q, p) T 1 (p, q) = (p, q + p ) finite orbits O (p,q) on rationals ã 2n [p, q ](iµ), with (p, q ) O (p,q), vector-valued modular form of weight 2 for the modular group PSL 2 (Z)

46 summing over orbits: ã 2n ( iµ; O(p,q) ) = (p,q ) O (p,q) ã 2n [p, q ](iµ) is an ordinary modular form of weight 2 for PSL 2 (Z) Question: which modular form is it? analyze zeros and poles structure to find ( out ) Example: for all n, modular form ã 2n iµ; O(0, 1 in 3 ) one-dimensional space spanned by G 14 (iµ) (iµ), with modular discriminant (cusp form weight 12) and G 14 is Eisenstein series weight 14 ( ) Example: for all n, modular form ã 2n iµ; O( 1 6, 5 in 6 ) one-dimensional space spanned by (iµ)g 6 (iµ) G 4 (iµ) 4